How to recover the target Q network's weights solely from the snapshots of the primary Q network's weights in DQN?
Let's add a step index to your expression
$$Q_{target}^{n} = (1-\tau)Q^{n-1}_{target} + \tau\, Q^{n-1}_{primary}$$
We can expand it one step further
$$Q_{target}^{n} = (1-\tau)^2Q^{n-2}_{target} + (1-\tau)\tau\, Q^{n-2}_{primary} + \tau\, Q^{n-1}_{primary}$$
And further
$$Q_{target}^{n} = (1-\tau)^3Q^{n-3}_{target} + (1-\tau)^2\tau\, Q^{n-3}_{primary} + (1-\tau)\tau\, Q^{n-2}_{primary} + \tau\, Q^{n-1}_{primary}$$
So, I guess, we can write a general formula for $m$ steps behind like:
$$Q_{target}^{n} = (1-\tau)^{n-m}Q^{n-m}_{target} + \tau\,\sum_{i=0}^{m-1} (1-\tau)^i Q^{n-i-1}_{primary} $$
For $n-m$ large enough $(1-\tau)^{n-m}$ should be close to 0 and you should be able to approximately reconstruct your $Q_{target}^n$ using only the history of $Q_{primary}$es
Edit: I've missed that you only have snapshots with some step between them. This is not ideal, but a possible way out would be to use, say, a linear interpolation between snapshot points.